Gravitational potential well

Let us use (12) to compute the gravitational redshift, the reduction in the frequency of waves as they climb out of a gravitational potential well. Recall that we obtained (12) by assuming gag is time independent. Imagine an oscillator (decaying atom, radar device) produces a wave train of sharp frequency u at a point xi. This means that N = u Ari is the number of cycles of the wave in an interval At of the (proper) time ticked by a clock at rest at xi. But by (12) we have the relation At = (c2 + 24w(xi))1/2At with the interval of t time spanned by the train. Thus the number of cycles can be written N = (c2 + 2(I> fxi))1 /2 At. Now the metric is not changing,... 

The existence of two stable steady states for the same values of the externally controlled constraints is known as bistability . A mechanical analogy to such a situation is the double well potential shown in Fig. 2. (No potential function is known to exist for open chemical systems that would play the role of the gravitational potential energy in... 

Other estimates of the amount and of the distribution of DM in the universe come from the study of large scale structures at more recent epochs in the cosmic evolution. The reason why cosmic structures contain a record of the DM distribution in the universe is due to the fact that the evolution of the parent density perturbations was dominated by their DM content from early times on (see Peacock in these Proceedings). Thus the study of galaxies and galaxy clusters - the largest gravitationally bound structures in the universe whose potential wells are dominated by DM - provide information on both the amount of DM and on its density distribution. 

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally... 

Many-body problems wnth RT potentials are notoriously difficult. It is well known that the Coulomb potential falls off so slowly with distance that mathematical difficulties can arise. The 4-k dependence of the integration volume element, combined with the RT dependence of the potential, produce ill-defined interaction integrals unless attractive and repulsive mteractions are properly combined. The classical or quantum treatment of ionic melts [17], many-body gravitational dynamics [18] and Madelung sums [19] for ionic crystals are all plagued by such difficulties. 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

One density-based method is the Dry Tribo-Separation process where separation relies not only on density, but also on the difference in friction coefficients of various particle sizes and shapes (Eiderman et al. 2000). Dry material is fed onto the surface of a rotating conical bowl and a combination of centrifugal, frictional, and gravitational forces separates the ash into two products one that is enriched in carbon and one that is carbon-lean. This process is currently under development but has demonstrated the potential to separate particles by size as well as by shape. 

---